Square-zero element
= Square-zero element
{title2=$a^2=0$}
An element of an <associative algebra> is square-zero if its square vanishes. Every such element is a <nilpotent element>. These elements are preserved by an <algebra isomorphism>, so their structure distinguishes algebras. Their set need not be a <vector space>: the matrix units $E_{12}$ and $E_{21}$ are square-zero, while their sum has square equal to the identity in $M_2(K)$. In a <square-zero ideal>, by contrast, every pairwise product vanishes and all its elements are square-zero. In $K[t]/(t^3)$ the square-zero elements are exactly the multiples of $t^2$, over every <field>: $(yt+zt^2)^2=y^2t^2$ forces $y=0$.